In the world of horse races, power functions like 2^{x} will always grow faster than plain old polynomials, no matter how high the degree of the polynomial. By "grow faster'' we mean that if we go far enough to the right on the graph, the power function will be on top of the polynomial. We also mean that

and

.

If a function has a power function term in it, we consider it a power function for now.

Now we'd better correct a little something: we could have a power function getting more and more negative instead of more positive. We could also have a polynomial getting more negative instead of positive. It's important to take signs into consideration when determining this limit, because we could have

## Practice:

What is | |

If we graph the functions *x*^{2} and 2^{x}, they intersect in three places: One of the points of intersection has a negative value of *x*, and two have positive values of *x*. After those intersections, as *x* approaches ∞ the graph of 2^{x} will always be on top of the function *x*^{2}. Here are some values of the two functions. We can see that when *x* = 2 and *x* = 4 the functions intersect, and that when *x* is greater than 4 the function 2^{x} is pulling away from the function *x*^{2}. It's like a horse race, where the function *x*^{2} is not having a good day. If we take these values and look at the quotient , here's what happens: We're already down to about 0.098 when *x* = 10, and we've barely started! When *x* = 100, we're down to Now that's small. We conclude that
| |

What is | |

We already know these functions intersect for some negative value of *x* and at *x* = 2 and *x* = 4. We also know that the function *x*^{2} is losing the horse race: for all *x* greater than 4, *x*^{2} will be below (or behind) 2^{x}. Taking the same values, we'll look at the new ratios now that we've turned the fraction over. These numbers don't seem to be approaching anything yet. Try something larger: We find that as *x* gets larger, the quotient will also get larger and larger without bound. Therefore
| |

Find the limit:

Answer

because the function on top is a power function (contains the term 3^{x}) and the function on the bottom is a polynomial.

Find the limit:

Answer

because the function on the top is a polynomial and the function on the bottom is a power function.

Find the limit:

Answer

because the numerator contains a power function with a negative coefficient, and the denominator a power function with a positive coefficient.